Ghana midfielder Michael Essien scored an own goal as a 4-1 defeat to Liverpool on Tuesday night ended Chelsea's hopes of a top-four finish.
Essien lasted the entire duration of the match and was booked after 44 minutes.
First Suarez produced a brilliant run from wide on the right to get to the byline and his cut-back was turned in by Essien in the 19th minute.
A slip from Terry put Henderson through on goal to curl a low shot past Ross Turnbull and then, in the 28th minute, Carroll headed down a corner for Agger to nod in from close range.
But a familiar foe returned again as Downing's 30-yard dipping volley struck the crossbar before the England winger drilled a penalty at the post late in the half after Ivanovic had been punished for knocking over Carroll.
Spot-kicks have also been an issue for the Reds with that the fifth out of six in the league which has gone begging this season, with others also missed in their runs to two cup finals.
Chelsea briefly got themselves back in the game five minutes after half-time when Florent Malouda's inswinging free-kick was turned in by Ramires five minutes after the break.
But the hosts did not back down and, if anything upped the pressure, which led to their fourth.
An out-of-position Turnbull mis-kicked straight into the path of Shelvey 30 yards out and he drilled a shot into an empty net.
Dalglish was criticised for leaving Carroll out of the starting line-up for the cup final despite the 23-year-old noticeably improving over the last six weeks in particular.